A polymer melt crystallises into spherulites with a constant nucleation rate and a…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Non-equilibrium statistical mechanics and stochastic thermodynamics

Soft Matter at the Thermal Scale, Ch Nucleation and the lifetime of a metastable phase

Problem

A polymer melt crystallises into spherulites with a constant nucleation rate I=1.0×1015 m−3 s−1I = 1.0\times10^{15}\,\mathrm{m^{-3}\,s^{-1}} and a constant radial growth rate G=1.0×10−8 m s−1G = 1.0\times10^{-8}\,\mathrm{m\,s^{-1}}, so the transformed fraction obeys the Avrami law for constant nucleation, X(t)=1−exp⁡ ⁣(−π3IG3t4)X(t) = 1-\exp\!\left(-\tfrac{\pi}{3}IG^3t^4\right). Find the half-crystallisation time t1/2t_{1/2}, defined by X(t1/2)=0.5X(t_{1/2}) = 0.5.

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